Éléments de statistique Partie I - Statistique fondamentaleetienne.roquain.free.fr/file/Slides_chap1.pdf · 2019. 9. 3. · Cadre formel Déﬁnition Une expérience statistique

Éléments de statistiquePartie I - Statistique fondamentale

Etienne Roquain(merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...)

Master 2 Probabilités et Finance

Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) 1 / 23

Chapitre 1. Modèle statistique

1 Données

2 Modèle statistique

3 Trois grands problèmes statistiques

Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) 2 / 23

1 Données



Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 3 / 23

Tirages pile ou face

0 200 400 600 800 1000

0.0

0.4

0.8

I Pièce équilibrée?I Nombre de tirages nécessaires?

Source : simulations en R


Caractéristiques corporelles

Taille Poids Tour de cheville

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0 100 200 300 400 500

1618

2022

2426

28

I Taille des femmes suit une loi gaussienne?I Prédiction poids à partir du tour de cheville?

Source : https ://ww2.amstat.org/publications/jse/v11n2/datasets.heinz.html


Caractéristiques appareils photos1000 appareils photos, 13 caractéristiques

I Prédiction prix en fonction des autres caractéristiques?I Prix lié à la dimension de l’appareil ?

Source : https ://www.kaggle.com/crawford/1000-cameras-datasetEtienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 6 / 23

Données qualitatives

Cheveux\Yeux Marron Bleu Noisette orangeBrun 68 20 15 5

Châtain 119 84 54 29Roux 26 17 14 14Blond 7 94 10 16

Yeux\Sexe Homme FemmeMarron 98 122

Bleu 101 114Noisette 47 46orange 33 31

I Couleur des yeux liée à la couleur des cheveux?

Source : R-package HairEyeColor


Mesures de polluants par laser

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8−

0.6

−0.

4−

0.2

0.0

I Courbe physique sous-jacente?

Source : R-package SemiParEtienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Données 8 / 23

Expression de gènes

gènesm

alad

esa

in

I Effets des gènes sur le cancer de la prostate?I Gènes impliqués dans le cancer de la prostate?

Source : R-package sda


Reconstitution de signal dimension 1

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0.0 0.2 0.4 0.6 0.8 1.0

−2

−1

01

I Débruitage?

Source : simulations en R


Reconstitution de signal dimension 2

I Débruitage?

Source : R-package EMD et simulations en R


1 Données



Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 12 / 23

Probabilité vs Statistique

I Soit (X1, . . . ,Xn) : (Ω,F)→ (X n,Xn) mesurable

I Probabilité : loi Pn de (X1, . . . ,Xn) donnée

(Ω,F ,P) −→ (X n,Xn,Pn)ω 7−→ (X1(ω), . . . ,Xn(ω)) = (x1, . . . , xn)

I Statistique : réalisation (x1, . . . , xn) ∈ X n , loi Pn ?

Modèle Pn ∈ P = {Pn,θ, θ ∈ Θ} lois candidates


Cadre formel

DéfinitionUne expérience statistique est définie par un triplet (X n,Xn,P) où (X ,X) estun espace mesurable et P est une famille de lois sur (X n,Xn).

Vocabulaire :I P est le modèle statistique.I L’entier n ≥ 1 est la taille de l’échantillon.I L’espace (X ,X) (ou (X n,Xn)) est l’espace d’observation (X = R où Rp).I Une paramétrisation du modèle P est une application θ ∈ Θ 7→ Pn,θ

satisfaisant P = {Pn,θ, θ ∈ Θ}.I L’ensemble Θ est l’espace des paramètres et θ est le paramètre du

modèle.Une observation (X1, . . . ,Xn) du modèle satisfait :

(Ω,F , {Pθ, θ ∈ Θ}) −→ (X n,Xn, {Pn,θ, θ ∈ Θ})ω 7−→ (X1(ω), . . . ,Xn(ω)) = (x1, . . . , xn)


Types de modèlesLe modèle P = {Pn,θ, θ ∈ Θ} (ainsi paramétré) est dit . . .

. . . identifiablesi ∀θ, θ′ ∈ Θ, Pn,θ = Pn,θ′ implique θ = θ′.

Sinon reconstruction du paramètre impossible à l’aide des données !

. . . produit

s’il existe {Pθ, θ ∈ Θ}, famille de lois sur (X ,X), tell que Pn,θ = P⊗nθ , θ ∈ Θ(c’est-à-dire X1, . . . ,Xn i.i.d.)

Convention : pour les modèles produits, la famille {Pθ, θ ∈ Θ} est le modèle.

. . . paramétrique

s’il existe p ≥ 1 tel que Θ ⊂ Rp, il est dit non-paramétrique sinon

. . . de grande dimension

si Θ ⊂ Rp, Vect(Θ) = Rp avec p « grand » (informel)Etienne Roquain (merci à Nathalie Akakpo, Lucien Birgé, Arnaud Guyader, Tabea Rebafka ...) Modèle statistique 15 / 23

Tirages pile ou face

0 200 400 600 800 1000

0.0

0.4

0.8

I Modèle {B(θ)⊗n, θ ∈ [0,1]}I Identifiable, produit, paramétrique.




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0 100 200 300 400 500

1618

2022

2426

28

I Par exemple modèle {N (µ, σ2)⊗n, θ = (µ, σ) ∈ R× (0,∞)}I Identifiable, produit, paramétriqueI Mais modèle {N (µ, σ2)⊗n, θ = (µ, σ) ∈ R2} pas identifiable.




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28

I Autre exemple modèle P = {P⊗n, P loi sur R }, θ = PI Identifiable, produit, non-paramétrique




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Documents

Éléments de statistique Partie I - Statistique fondamentaleetienne.roquain.free.fr/file/Slides_chap1.pdf · 2019. 9. 3. · Cadre formel Déﬁnition Une expérience statistique